Quantifying environmental and other detrimental risks from exposure to hazardous agents is an important component in the process of risk evaluation and assessment. The primary goal of this project is to extend and study statistical confidence bands for use in low-dose risk extrapolation. Application is directed to environmental and occupational risk assessment studies where human or animal data are used to set benchmark or other safe low-dose levels of a hazardous agent, but where study information is limited to high dose levels of the agent. Methods are considered for estimating upper confidence limits on predicted risk for various endpoints measured on both continuous and discrete scales. From the simultaneous confidence bounds, lower confidence limits on the benchmark dose (BMD) associated with a particular risk are calculated. An important feature of the simultaneous construction is that any inferences based on inverting the simultaneous confidence bounds apply automatically to inverse bounds on the BMD. The methodology extends previous results for simultaneous confidence bands to normally and non-normally distributed data with dose-dependent variance models, and to higher-dimensional parameter spaces associated with non-linear dose-response functions. An evaluation phase of the project studies the small-sample operating characteristics of the new methodology via Monte Carlo computer calculations, and applies the new methods to existing data for a number of endpoints encountered in quantitative risk assessment/low-dose extrapolation problems. The new methods fill existing gaps in low-dose risk extrapolation, and have application to a wide variety of data-analytic scenarios in quantitative risk assessment.